Discretization of Maxwell’s Equations for Non-inertial Observers Using Space-Time Algebra

• Published in: Advances in applied Clifford algebras Vol. 28; no. 1; pp. 1 - 43
• Main Authors: Klimek, Mariusz, Kurz, Stefan, Schöps, Sebastian, Weiland, Thomas
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Coding; Discretization; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Maxwell's equations; Mesh generation; Physics; Physics and Astronomy; Relativism; Relativistic effects; Spacetime; Theoretical; Three dimensional motion; Time setting; Clifford’s geometric algebra; Minkowski space-time; Finite integration technique; 65M06; 15A66; 65M60; Whitney finite elements
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-018-0841-3

We employ classical Maxwell’s equations formulated in space-time algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac’s effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework.